Gδ -sets in topological spaces and games

Just, Winfried; Scheepers, Marion; Steprans, Juris; Szeptycki, Paul

Just, Winfried ; Scheepers, Marion ; Steprans, Juris ; Szeptycki, Paul

Fundamenta Mathematicae, Tome 154 (1997), p. 41-58 / Harvested from The Polish Digital Mathematics Library

Résumé

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_{n}$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_{n}$ of X. The players must obey the rule that $O_{n} \subseteq O_{n + 1} \subseteq T_{n + 1} \subseteq T_{n}$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a $G_{δ}$ -set. To what extent is the converse true? We show that: (A) For ℱ the collection of countable subsets of X: 1. There are subsets of the real line for which neither player has a winning strategy in this game. 2. The statement “If X is a set of real numbers, then ONE does not have a winning strategy if, and only if, every countable subset of X is a $G_{δ}$ -set” is independent of the axioms of classical mathematics. 3. There are spaces whose countable subsets are $G_{δ}$ -sets, and yet ONE has a winning strategy in this game. 4. For a hereditarily Lindelöf space X, TWO has a winning strategy if, and only if, X is countable. (B) For ℱ the collection of $G_{σ}$ -subsets of a subset X of the real line the determinacy of this game is independent of ZFC.

Publié le : 1997-01-01

Zbl 0880.90158

EUDML-ID : urn:eudml:doc:212214

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     author = {Winfried Just and Marion Scheepers and Juris Steprans and Paul Szeptycki},
     title = {G$\delta$ -sets in topological spaces and games},
     journal = {Fundamenta Mathematicae},
     volume = {154},
     year = {1997},
     pages = {41-58},
     zbl = {0880.90158},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv153i1p41bwm}
}

Just, Winfried; Scheepers, Marion; Steprans, Juris; Szeptycki, Paul. Gδ -sets in topological spaces and games. Fundamenta Mathematicae, Tome 154 (1997) pp. 41-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv153i1p41bwm/

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